Physics 777 Plasma Physics and Magnetohydrodynamics (MHD) Instructor: Gregory Fleishman Lecture 9. Large-Amplitude Waves and Discontinuities 28 October 2008

Plan of the Lecture Simple Waves Nonlinear Waves, Solitons Discontinuities in the Plasma Shock Waves Strong Turbulence

Section 1. Simple Waves

Set of MHD equations reads

& If If Section 1.1. Entropy simple waves - arbitrary For other simple waves where

Section 1.2. Alfven simple waves &

If for some then for for all Section 1.3. Fast and slow simple waves Consider simple case

Section 2. Nonlinear Waves, Solitons Korteweg – de Vries – Burgers Equation, which includes nonlinearity, dispersion, and dissipation as additive terms: where Burgers Equation Korteweg – de Vries (KdV) Equation

Section 2.1. Solitons Consider KdV Equation Seek solution

Necessary conditions: Case I For or result in

Rarefaction! Case II Jakobi elliptic function; - Cnoidal wave Chain of solitons

Section 3. Discontinuities and Shocks in the Plasma

Upstream Downstream

Section 4. Evolutionarity of the Discontinuities

Section 5. Strong Turbulence

Upstream Downstream & & &

Section 6. Homework Complete Progress Reports on the research project. Include Introduction, the work done with plots, and plan for the remaining work. Be ready for brief presentation of the Progress Report.